Residue perturbation (RP) is often used as a means for enforcing passivity of models describing the linear properties of electrical components. One known RP approach uses quadratic programming (QP) for solving a least squares problem with constraints.
As an example, consider a pole-residue model of an admittance matrix Y.
                                          Y            ⁡                          (              s              )                                =                                                    ∑                                  m                  =                  1                                N                            ⁢                                                R                  m                                                  s                  -                                      a                    m                                                                        +            D                          ,                            (        1        )            wherein s is the angular frequency, Rm with m=1 to N are matrices independent of s (with N being the number of poles or resonances taken into account, N<∞), D is a matrix independent of s, and am with m=1 to N are the complex angular frequencies of the poles or resonances.
The model parameters are to be perturbed so that the perturbed model satisfies the passivity criterion that the real part of the eigenvalues of Y is positive for all frequencies, i.e.
                              eig          (                      Re            ⁢                          {                              Y                +                                                      ∑                                          m                      =                      1                                        N                                    ⁢                                                            Δ                      ⁢                                                                                          ⁢                                              R                        m                                                                                    s                      -                                              a                        m                                                                                            +                                  Δ                  ⁢                                                                          ⁢                  D                                            }                                )                >        0                            (                  2          ⁢                                          ⁢          a                )            
The perturbation is to be done so as to minimize the change to the original model, i.e.
                              Δ          ⁢                                          ⁢          Y                =                                                            ∑                                  m                  =                  1                                N                            ⁢                                                Δ                  ⁢                                                                          ⁢                                      R                    m                                                                    s                  -                                      a                    m                                                                        +                          Δ              ⁢                                                          ⁢              D                                ≅          0                                    (                  2          ⁢                                          ⁢          b                )            The known way of handling equation (2b) is to minimize the change to ΔY in the least squares sense.